Extensions of certain classical integrals of Erdélyi for Gauss hypergeometric functions

On Certain Integrals Related to Saran’s Hypergeometric Function FK

“…Equation ( 2) was first derived by Erdélyi [17] by making use of the fractional integration by parts and was later rediscovered by Joshi and Vyas [21] by using the series manipulation techniques. Additionally, Equation (2) has some important applications.…”

“…Let S denote the triple series in (21). By summing it over m, n and p (as in the proof of Lemma 2), we obtain…”

q-Extensions of certain Erdélyi type integrals

“…Several other well-known special cases of the q-integrals (1.2)-(1.14) can be discussed on the line of their ordinary versions studied in [6]. Further, the q-integrals (1.2)-(1.13) generalize certain q-hypergeometric transformations and also give rise to some hitherto unrecorded q-hypergeometric transformations.…”

“…In a recent paper [6], we gave an alternative way of proof for Erdélyi's integrals (see [6, Eqs. (1.3)-(1.5)]) using series manipulation technique and classical summation theorems.…”

“…Now applying the triple series manipulation[6, p. 128], on this triple q-series and summing the inner n-series by the q-Chu-Vandermonde sum [4, p. 236, (II.6)] and then the inner m-series by the q-PfaffSaalschütz sum[4, p. 237, (II.12)], we get the LHS of (1.2) and this completes the proof. The (1.3) can be proved on the line of its ordinary analogue[6, Eq. (4.1)].…”